Type Indeterminacy: A Model of the KT(Kahneman-Tversky)-man

Transcription

1 Type Indeterminacy: A Model of the KT(Kahneman-Tversky)-man Ariane Lambert Mogiliansky, Shmuel Zamir and Hervé Zwirn October 10, 2003 Abstract In this note we propose to use the mathematical formalism of Quantum Mechanics to capture the idea that agents preferences, in addition to being typically uncertain, can also be indeterminate. They are determined (realized, and not merely revealed) only when the action takes place. An agent is described by a state which is a superposition of potential types (or preferences or behaviors). This superposed state is projected (or collapses ) onto one of the possible behaviors at the time of the interaction. In addition to the main goal of modelling uncertainty of preferences which is not due to lack of information, this formalism, seems to be adequate to describe widely observed phenomena like framing and instances of noncommutativity in patterns of behavior. We propose two experiments to test the theory. JEL: D80, C65, B41 CERAS, Ecole Nationale des Ponts et Chaussées URA 2036, Paris ariane.lambert@mail.enpc.fr CNRS-EUREQua, CREST-LEI Paris and, Center for the Study of Rationality, Hebrew University, zamir@math.huji.ac.il IHPST, UMR 8590 CNRS/Paris 1, and CMLA, UMR 8536 CNRS/ENS Cachan hzwirn@m4x.org 1

2 1 Introduction Recently, models of quantum games have been proposed to study in which respect the extension of classical moves to quantum ones (i.e. complex linear combination of classical moves) could affect the analysis of the game. For example, it has been shown (Eisert et al. 1999) that allowing the players to use quantum strategies in the Prisoners Dilemma, is a way to escape the well-known bad feature of this game. 1 From a game theoretical point of view the approach consists of changing the strategy spaces and thus the interest of the results rests on the appeal of these changes. 2 This paper also proposes to use the mathematical formalism of Quantum Mechanics but with a different intention; the goal being to model uncertain preferences. The basic idea is that the Hilbert space model of Quantum Mechanics can be thought of as a very general contextual predictive tool particularly well suited to describe experiments in psychology or in revealing preferences. The well established Bayesian approach suggested by Harsanyi to model incomplete information consists of a chance move that selects the actual types of the players and informs each player of his own type. For the purpose of this note, we underline the following essential implication of this approach: all uncertainty about a player s type reflects exclusively others incomplete knowledge about it. This follows from the fact that a Harsanyi type is fully determined. It is a well-defined characteristic of a player which is known to him. Consequently, from the point of view of other players, uncertainty about the type can only be due to lack of information. Each player has a probability distribution over the type of the other players, but his own type is fully determined and is known to him. This brings us to the firstimportantpointatwhichwedepartfromthe classical approach: we propose that in addition to informational reasons, the uncertainty concerning preferences is due to indeterminacy: prior to the moment a player acts, his (behavior) type is indeterminate. The state, representing the player, is a superposition of potential types. It is only at 1 In the classical version of the dilemma, the dominant strategy for both players is to defect and thereby to do worse than if they would both decide to cooperate. In the quantum version, there is a couple of quantum strategies which is both a Nash equilibrium and Pareto optimal and whose payoff is that of joint cooperation. 2 This approach is closely related to quantum computing. It makes use of the notion of q-bits (quantum units of information) and of that of measurement. 2

3 the moment when the player selects a choice, that a specific typeisrealized. It is not merely revealed but rather determined, in the sense that prior to the choice, there is an irreducible richness of potential types. Thus we suggest that in modelling a decision situation, we do not assume that the preferences characteristics can always be fully known with certainty (not to the decision maker and even not to the analyst). Instead, what is known is the state of the agent: a vector in a Hilbert space spanned by types which encapsulates all idiosyncratic information needed to predict how the agent is expected to behave in different decision situations. This idea, daringly borrowed from Quantum Mechanics to the context of decision and game theory, is very much in line with Tversky and Simonson according to which There is a growing body of evidence that supports an alternative conception according to which preferences are often constructed - not merely revealed - in the elicitation process. These constructions are contingent on the framing of the problem, the method of elicitation, and the context of the choice. (in Kahneman and Tversky 2000). This view is also consistent with that of cognitive psychology that teaches to distinguish objective reality from the proximal stimulus to which the observer is exposed, and to distinguish both reality and the stimulus from the mental representation of the situation that the observer eventually constructs. More generally this view fits with the observation that players (even highly rational) may act differently in game theoretically equivalent situations which differ only by seemingly irrelevant aspects (framing, prior unrelated events etc.). Our theory for why people act differently in game theoretical equivalent situations is that they may not be representable by the same state: (revealed) preferences are contextual because of intrinsic indeterminacy. The basic analogy with Physics, which makes it appealing to borrow the mathematical formalism of Quantum Mechanics to social sciences, is the following: We view observed play, decisions and choices as something similar to the result of a measurement (a measurement of the player s state of preferences or the player s type). The decision situation is modeled as an operator (called observable), and the resulting behavior as an eigenvalue of that operator. The analogy to the non-commutativity of observables (a very central feature of Quantum Mechanics) is, in many empirically observed phenomena, like the following one. Suppose that we subject a given agent to the same decision situation in two different contexts (the contexts may vary with respect to the decision situations that precede the investigated one, or it canvarywithrespecttotheframinginthepresentationofthedecision,(cf. 3

4 Selten 1998)). If we do not observe the same decision in the two contexts, then the classical approach is to say that the two decision situations are not the same; they should be modeled so as to incorporate the context. In many cases however such an assumption i.e. that the situations are different, is difficult to justify. And so the standard theory leaves a host of behavioral phenomena unexplained: the behavioral anomalies (cf. McFadden 1999). In contrast, we propose to say that the observed decisions were not taken by an agent of the same type. The context e.g. a past decision, is viewed as an operator that does not commute with the operator associated with the investigated decision situation; its operation on the player has changed the state (the type) of the agent. As in Quantum Mechanics the phenomenon of non-commutativity of decision situations (measurements) leads us to make a hypothesis that an agent s preferences or a preference system is represented by a state which is indeterminate and gets determined (with respect to any particular type characteristics) in the process of interaction with the environment. The objective of this paper is to propose a theoretical framework that extends the Bayes-Harsanyi model to accommodate various forms of the socalled behavioral anomalies. Adopting McFadden s terminology, we attempt to provide a model for the KT(Kahneman-Tversky)-man as opposed to what he calls the Chicago man (McFadden 1999). Our work is related to Random Utility Models (RUM) as well as to Behavioral economics. RUM models have provided very useful tools for explaining and predicting deviations from standard utility models. RUM, however cannot accommodate the kind of drastic and systematic deviations characteristic of the KT man. These models rest on an hypothesis of consumer sovereignty i.e. the primacy of desirability over availability, and they assume stable taste templates (see McFadden 2000 for a very good survey). In RUM preferences are non-contextual by construction while the proposed Hilbert Space Model (HSM) of preferences is a model of contextual preferences. Behavioral economics have contributed a variety of explanations for deviations from self-interest (see Camerer 1997). Most often they elaborate on the preference function itself introducing for instance trade off contrast, extremeness aversion (Kahneman and Tversky ed. 2000), fairness concerns (Rabin 1993) or self-control costs (Gul and Pesendorfer 2001) 3. The explanations proposed by behavioral economics are 3 Gul and Pesendorfer s work also addresses deep conceptual issues related to the specific class of behavioral anomalies they study. 4

5 often specific to the deviation that is being explained. Yet, other explanations would appeal to bounded rationality e.g. superficial reasoning (Selten 1998). In contrast, the HSM is a framework model that addresses structural properties of preferences. In section 2, the framework is presented and basic notions of quantum theory are introduced. In Section 3, we propose applications to Social Sciences. We compare the predictions and interpretations of the HSM with those of probabilistic models. Two experiments to test the theory are suggested. An application of the approach to framing effects is proposed. Concluding remarks are gathered in the last section. 2 The basic framework In this section we present the basic notions for our framework. They are heavily inspired by the mathematical formalism of Quantum Mechanics ( see e.g. Cohen-Tannundji, Dui and Laloe,1973) from which we also borrow the notation. 2.1 The notion of state The object of our investigation is the choice behavior which we interpret as revelation of the preferences of an agent in what we call a Decision Situation (DS). In this paper we focus on non-repeated non-strategic decision situations. This can be the choice between buying a Toshiba laptop at price x or a Compaq laptop at price y. The choice between a sure gain of $100 or a bet with probability 0.5 to win $200 and 0.5 to win $0, is a DS as well as the choice between investing in a project or not, etc. When considering games we review them as decision situations from the point of view of a single player 4. An agent or a preference system is represented by a state which captures all the agent s potential behavior in the decision situations under consideration. Mathematically, a state ψi is a vector in a Hilbert space H over the field of the complex numbers C. This space has to be large enough to accommodate all the decision situations we want to study. The relationship between H and the decision situations will be specified later. 4 All information (beliefs) and strategic considerations are embedded in the definition of the choices. Thus the agent s play of C is a play of C given e.g. his information (knowledge) about the opponent. 5

6 For technical reasons related to the probabilistic content of the state, it has to be of length one that is, hψ ψi 2 =1. So all vectors of the form λ ψi where λ²c correspond to the same state which we represent by such vector of length one. For this reason we shall consider only linear combinations in H of the form ψi = P n i=1 λ i ϕ i i with P n i=1 λ i 2 =1(such a linear combination is called a superposition). If ϕ 1 i... ϕ n i are all states (i.e. unit vectors ) then the superposition ψi is also a state. 2.2 The notions of observable and measurement A decision situation is defined by the set of alternative choices available to the agent. When an agent selects a choice out of a set corresponding to a decision situation, we say that he plays the DS. Each choice corresponds to a type in the relevant DS. To every Decision Situation A, weassociateanobservable, namely a specific Hermitian operator on H which, for notational simplicity, we denote also by A. 5 If A is the only decision situation we consider, we can assume that its eigenvectors which we denote by 1 A i, 2 A i,..., n A i all correspond to different eigenvalues, denoted by 1 A, 2 A,...,n A respectively. A k A i = k A k A i, k =1,..., n. As A is Hermitian, there is an orthonormal basis of the relevant Hilbert space H formed with its eigenvectors. The basis { 1 A i, 2 A i,..., n A i} is the unique orthonormal basis of H consisting of eigenvectors of A. It is thus possible to represent the agent s state as a superposition of the vectors of this basis: ψi = nx λ k k A i, (1) k=1 P where λ k C, k {1,..., n} and n λ k 2 =1. k=1 The expansion (1) can also be written as H = H 1A,..., H na (2) where denotes the direct sum of the subspaces H 1A,..., H na spanned by 1 A i,..., n A i respectively. 6 Or, in terms of the projection operators 5 An operator A on H is Hermitian if A 0 = A where A 0 is the adjoint (Hermitian conjugate) defined by hϕ A 0 ψi = hψ A ϕi for all ϕ and ψ in H. 6 That is, for i 6= j any vector in H ia is orthogonal to any vector inh ja and any vector in H is a sum of n vectors, one in each component space. 6

7 P 1A +,..., +P na = I H where P ia is the projection operators on H ia and I H is the identity operator on H. This presentation will prove to be convenient when treating more than one observable. In fact if we want to treat both observables A with eigenvalues {1 A, 2 A,..., n A },and B with eigenvalues {1 B, 2 B,...,m B }, we express the Hilbert space H as a direct sum in two different ways: H = H 1A,..., H na and H = H 1B,..., H mb where H ia is the subspace of eigenvectors of A with eigenvalue i A (similarly of H jb ). Note however that in this case H ia and H jb will not in general be one dimensional. In the general case, when the subspaces H ia are not one dimensional (i.e. when the eigenvalues i A are degenerated) the superposition in eq. (1) still exists and is unique for each ψi but the vectors k A i ²H ka, may depend on ψi. 7 The realization of a type for an agent in the state ψi, in a decision situation A is the actual choice of a particular action among all the possible actions in that situation. It is represented by a measurement of the observable A associated with this decision situation. According to the so-called Reduction Principle (see appendix), the result of such a measurement can only be one of the n eigenvalues of A. If the result is m A i.e. the player selected the choice m A, the superposition P λ i i A i collapses onto the eigenvector associated with the eigenvalue m A (i.e. the initial state ψi is projected onto the the subspace H ma of eigenvectors of A with eigenvalue m A ). The probability that the measurement yields the result m A is equal to hm A ψi 2 = λ m 2 (where h i denotes the inner product in H) 8. As usual, we will interpret the probability of m A either as the probability that one agent in the state ψi selects the choice m A or as the proportion of the agents who will make the choice m A in a population of many agents, all in the state ψi. Remark: Clearly, when only one DS is considered, the above description is equivalent to the traditional probabilistic representation of an agent by a probability vector (α 1,..., α n ) in which α k is the probability that the agent will choose action k A. A distinction however is that in the HSM the probability 7 To see that such expansion exists and is unique take a basis of H which consists of basis of the component subspaces H ka,expand ψi in this basis and let k A i be the part of the sum in H ka of that expansion. This k A i is uniquely defined for each ψi. 8 We here for simplicity assume that all eigenvalues are non degenerated. We return to that issue below. 7

8 distribution is not directly given by the coefficients of the linear combination but by the square of their module. The advantage of the proposed formalism is in studying several decision situations and the interaction between them. 2.3 More than one Decision Situation When studying more than one DS, say A and B, it turns out that a key question is whether the corresponding observables are commuting operators in H i.e. whether AB = BA. The question whether two DSs can or cannot be represented by two commuting operators is an empirical question on which we comment later, we first study the mathematical implications of this feature Commuting Decision Situations Let A and B be two DSs. If the corresponding observables commute then there is an orthonormal basis of the relevant Hilbert space H formed by eigenvectors common to both A and B. Denoteby ii (for i =1,..,n) these basis vectors. We have A ii = i A ii and B ii = i B ii. In general the eigenvalues can be degenerated ( i.e. for some i and j, i A = j A or i B = j B ). Any normalized vector ψi of H canbewritteninthisbasis: ψi = X i λ i ii where λ i ²C, and P i λ i 2 =1. If we measure A first, we observe eigenvalue i A with probability: p A (i A )= X λ j 2 (3) j;j A =i A If we measure B first, we observe eigenvalue j B with probability: p B (j B )= X λ k 2 (4) k;k B =j B After having measured B and obtained the result j B, the state ψi is projected into the eigensubspace E jb spanned by the eigenvectors of B associated with j B. More specifically, it collapses onto the state: 8

9 (the factor 1 q P k;kb =jb λ k 2 ψjb = 1 q P k;kb=jb λ k 2 X k;k B =j B λ k ki is necessary to make ψjb aunitvector). Now when we measure A on the agent in the state ψjb, we obtain ia with probability 1 X 2 p A (i A j B )= Pk;k B =j B λ k 2 λ k k;k B =j B and k A =i A So when measuring B first and then A, the probability to observe the eigenvalue i A is: p AB (i A ) = P j p B (j B ) p A (i A j B ) = P P P 1 j k;k B =j λ k 2 k;k B =j B λ k 2 P l;l B =j B λ l 2 B and l A =i A = P P j l;l B =j B λ l 2 = P l;l A =j A λ l 2 and l A =i A = p A (i A ) Hence, p AB (i A )=p A (i A ), i, and similarly p BA (j B )=p B (j B ), j. This expresses the fact that there is no interference between the two observations: we have the same probability distribution on the result of the measurement of A whether we measure A directly or measure B first and then measure A. The measurement of B does not perturb the measurement of A. In this case it is meaningful to speak of measuring A and B simultaneously. Whether we measure first A and then B or first B and then A, wehave P the same probability distribution on the joint outcome namely p (i A j B )= λ k 2. In other words (A, B) have definite pair of values (i A, j B ) k;k B =j B and k A =i A is a well defined event. Remark: Note that again, as in the case of one DS, for such two DSs our model is equivalent to a standard (discrete) probability space in which the elementary events are {(i A,j B )} and p (i A j B )= P k;k B =j B λ k 2. In and k A =i A particular, in accordance with the calculus of probability we see that the conditional probability formula holds: p AB (i A j B )=p A (i A ) p B (j B i A ). 9

10 As an illustration of an example of this kind, consider the following two decision problems: Let A be the decision situation of choosing between a week vacation in Tunisia and a week vacation in Italy. And let B be the choice between buying 1000 euros of shares in Bouygues telecom or in Deutsche telecom. If the corresponding observables commute, we expect that a decision on the portfolio (B) does not affect decision-making regarding the location for vacation (A). And thus the order in which the decisions are made does not matter. It is quite plausible that A and B commute but whether or not this is in fact the case, is of course an empirical question. Note that the commutativity of the observables does not exclude statistical correlations between the two observations. To see this consider the following example in which A and B each have two degenerated eigenvalues in a four dimensional Hilbert space. Denote i A j B i (i =1, 2, j =1, 2) the eigenvector associated with eigenvalues i A of A and j B of B and let the state ψi be given by: q A1 B i + ψi = Then p A (i A 1 B )= q q q A2 B i A1 B i A2 B i = 3 (for i =1or i =2) 4 So if we first measure B and find say 1 B, it is much more likely (prob. 3 4 ) that measuring A we find 1 A rather than 2 A (prob. 1 ). In other words the statistical correlation is informational: information about one of the observable 4 is relevant to the prediction of the value of the other. But the two interactions (measurements), do not perturb each other i.e., the distribution of the outcomes of the measurement of A isthesamewhetherornotwemeasure B first Non commuting Decision Situations It is when considering Decision Situations associated with observables that do not commute that the predictions of the HSM differ from those of the probabilistic model. As we shall see, the distinction is due to the presence of cross terms also called interferences. In the appendix we provide a brief description of the classical interferences experiment in Physics. For the sake of simplicity, assume that two decision situations A and B have the same number n of possible choices, that means that the observables 10

11 A and B have the (non degenerated) eigenvalues 1 A, 2 A,..., n A and 1 B, 2 B,..., n B respectively and each of the sets of eigenvectors { 1 A i, 2 A i,..., n A i} and { 1 B i, 2 B i,..., n B i} is an orthonormal basis of the relevant Hilbert space. So each element of one basis, say j B i, is a superposition of the elements of the other basis: j B i = nx µ ij i A i i=1 Let ψi be the initial state of the player. ψi = nx λ j j B i = j=1 nx j=1 i=1 nx λ j µ ij i A i If the agent plays directly A, he will choose i A with the probability 2 np p A (i A )= j µ. If he plays first B, he will select choice j j=1λ ij B with the probability λ j 2 andwillbeprojectedinthestate j B i then he will select choice i A in the game A with the probability µij 2. So the probability of P choice i A after having played B is p AB (i A )= n λ j 2 µij 2 whichisingeneral different from j µ. That is: Playing B first, changes the way A j=1 2 np j=1λ ij is played. The difference stems from the so called interference terms: p A (i A ) = 2 nx nx λ j µ = λ ij j 2 µij 2 X +2 Re λj 0µ ij λ jµ ij 0 j=1 j=1 j6=j 0 {z } Interference term = p AB (i A )+interference term The interference term is the sum of cross terms involving the coefficients of the superposition which are also called the amplitudes (of the probabilities). Some helpful intuition about interference effects may be provided using the concept of propensity ( K. Popper s (1992)). Imagine an agent s mind as a system composed of a large variety of propensities to act (corresponding 11

12 to the different possible actions). As long as the agent is not required to choose his action, propensities coexist in his mind; The agent has not made up his mind. A decision situation forces him to determine himself. The DS operates on this state of hesitation where the alternative propensities are present. The DS triggers the emergence of a single type (of behavior). The agent s state of hesitation is replaced by a determinate preference. The cross terms express competition/reinforcement of propensities in this process, and they contribute to determining the probabilities for observing the emergence of the realized type. An illustration of an example of this kind may be supplied by the experiment reported in Knez and Camerer (2000). The two DSs studied are the Weak link coordination (WL) game and the Prisoner s Dilemma (PD) game. 9 They compare the distribution of choices in the Prisoner Dilemma (PD)gamewhenitisprecededbyaWeakLink(WL)gameandwhenonly the PD game is being played. Their results show that playing the WL game affects the play of individuals in the PD game. The explanation offered by the authors makes use of what they call a transfer effect. 10 The precedent hypothesis is about shared expectations of cooperative play. This informational argument cannot explain the high rate of cooperation (37.5 %) in the last round of the PD game (table 5 p. 206) however. Instead, the experimental results are consistent with the hypothesis that the play of the WL game affected players preferences. We propose that the WL and the PD dilemma are two DS that do not commute. In such a case we do expect a difference in the distributions of choices in the PD depending on whether or not it was preceded by a play of the WL or another PD game. This distinction is due to interference effects (present in the second case) which is a feature of behavior indeterminacy. Remark: In this case, A and B cannot have simultaneously a defined 9 The weak link game is a type of coordination game in which each player picks an action from a set of integers. The payoffs aredefined in such a manner that each player wants to select the minimum of the other players but everyone wants that minimum to be as high as possible. 10 The transfer effect hypothesis is as follows: the shared experience of playing the efficient equilibrium in the WL game creates a precedent of efficient play strong enough to (...) lead to cooperation in a finitely repeated PD game, p.206, Knez and Camerer (2000). 12

13 value and there is no probability distribution on the event to have the value i A for A and the value j B for B. The conditional probability formula does not hold any more. p A (i A ) 6= nx p B (j B ) p(i A j B ) j=1 2 np P Indeed, j µ = p A (i A ) 6= j=1λ n P p B (j B ) p(i A j B )= n λ j ij 2 µij 2. j=1 j=1 In terms of our example above, the joint probability for event (i A,j B ) which is to play i A inthewlgameandj B inthepdcannotbedefined in a single probability space. Playing first the WL game does affect the distribution of observed play in the PD. 3 Indeterminacy in Social Sciences 3.1 Context dependent preferences As a first application let us consider a situation when a DS is played in a context. The context is an interaction that precedes the investigated DS. 11 Specifically consider the decision operator associated with the Prisoner s Dilemma (PD) A with { 1 A i, 2 A i} as eigenvectors corresponding to the choices cooperate (C) associated with eigenvalue 1 A and defect (D) associated with eigenvalue 2 A.Thepayoffsaregiveninthematrixbelowwhere the choices of the second player are put into brackets to indicate that we are considering only one agent (the row player) as a decision maker while the other (the column player) is part of the description of the DS. 1 A (C) 2 A (D) [C] Prisoner s Dilemma [D] What we call context here corresponds to what Tversky and Simonson call background context p.521 in Kahneman and Tversky (2002). 13

14 It is possible to write ψi in this basis: ψi = λ 1 1 A i + λ 2 2 A i Suppose that before playing the PD the agent has played another DS, a dictator game, DG, 12 (with two options) associated with an operator B that has 1 B i, 2 B i as its eigenvectors corresponding to the actions share fairly (F) and share selfishly (S). The dictator game is played against a third player i.e. not involved in the PD game. For me For him 1 B (F ) B (S) 95 5 Dictator Game It is also possible to write ψi in this basis: ψi = ν 1 1 B i + ν 2 2 B i. Assume that the two operators do not commute. We write the eigenvectors { 1 B i, 2 B i} as superpositions of the basis { 1 A i, 2 A i} : 1 B i = µ 11 1 A i + µ 12 2 A i 2 B i = µ 21 1 A i + µ 22 2 A i Hence: ψi = λ 1 1 A i + λ 2 2 A i = ν 1 1 B i + ν 2 2 B i (5) = (ν 1 µ 11 + ν 2 µ 21 ) 1 A i +(ν 1 µ 12 + ν 2 µ 22 ) 2 A i (6) The expression in (5) and (6) illustrates a central feature of the HSM formalism. Namely that any two DS associated with observables that do not commute, are linked by a basis transformation matrix S with S ji = hi A j B i = µ ji. In our two dimensional example S writes µ h1a 1 S = B i h2 A 1 B i h1 A 2 B i h2 A 2 B i so we have 12 The Dictator Game is a two players game in which one player, the Dictator, decides on a split of an amount of money, say 100 units, between him and the other player who is actually a dummy. 14

15 µ 1A i ψi =(λ 1 λ 2 ) 2 A i =(λ 1 λ 2 ) S µ 1B i 2 B i The basis transformation matrix that links any two non-commuting observables is of course independent of ψi. The elements of this matrix do have an interpretation as (the square root of) the (statistical) correlations between the types of the two DSs. These invariants can, in principle, be estimated using empirical data and used to calibrate predictive models of decision behavior (cf experiment 2 below). It follows from (6) that if the agent directly plays the PD, he will play the action i A with the probability p A (i A )= (ν 1 µ 1i + ν 2 µ 2i ) 2. That means that both potential types (fair (1 B ),selfish(2 B ))affect his behavior. Playing the dictator game corresponds to the collapse of ψi onto 1 B i or 2 B i. The probability for a collapse on k B i is ν k 2 and the probability that an agent of type b k i plays the action i A is µ ki 2. Thus, when playing B first, the agent will play the action i A with the probability p AB (i A )= ν 1 2 µ 1i 2 + ν 2 2 µ 2i 2 6= p A (i A ). The probability for playing action i A is not the same as when the agent does not play dictator game first. The reader could remark that it seems possible to obtain the same results in the standard probabilistic representation through an appropriate modification of the probability space. The next section addresses this question. Basically the argument is that in applications to Social Sciences when the explanation of empirical data calls for assuming changes in preferences for no good (plausible) reason, it may be more appropriate to model preferences as being indeterminate. The HSM then allows to make predictions that can be empirically tested. Two experiments are briefly sketched. 3.2 HSM and the classic probabilistic model This section is intended to clarify the relationship between the proposed Hilbert space representation and the classical probabilistic representation. Our typical setup is that of two observables, A with eigenvalues {1 A, 2 A,..., n A }, and B with eigenvalues {1 B, 2 B,..., m B }. So to treat both observables we express the Hilbert space H as a direct sum in two different ways:. H = H 1A,..., H na and H = H 1B,..., H mb where H ia is the subspace of eigenvectors of A with eigenvalue i A (similarly 15

16 of H jb ). Given the state ψi H of the agent under consideration, it can be represented as Similarly, ψi = X i ψi = X j λ i i Aψ i, ν j j Bψ i, X λ i 2 =1, i Aψ i H ia,i=1,..., n (7) i X ν j 2 =1 j Bψ i H jb,j=1,..., m (8) i Now, states j Bψ i H jb of eq. (8) can also be written as a superposition of the form (7) j Bψ i = X X µ ij i AjB i, µij 2 =1 iajb i H ia,i=1,..., n (9) i i Note that the vectors i Aψ i and i AjB i,beingbothinh ia,areeigenvectorof A with the same eigenvalue i A. The non commutativity of A and B is the violation of the equation p A (i A )=p AB (i A ), i =1,...n (10) which is λ i 2 = P µij 2 j νj 2. That is, the probability distribution on the outcomes of the measurement A may be affected by the fact that we measured B first and then A. Can this be modelled in a standard probability theory? The answer is obviously yes. To do that let A be a random variable with values in {1 A, 2 A,..., n A, } with probabilities p A (i A )=α i λ i 2,i=1,..., n which will be the equivalent of eq. (7). Similarly the equivalent of eq. (8) is obtained if B is a random variable with values in {1 B, 2 B,..., m B } and with probabilities p B (j B )=β j ν j 2,j=1,...,m. What is the analogue of eq. (9)? For each value j B of B there is a probability distribution p AjB of the random variable A given by p AjB (i A )=γ ij µij 2 ; i =1,...n, j =1,..., m (11) Then, when measuring B and then A, the distribution of outcomes is given by p AB (i A )= X β j γ ij i =1,..., n. j 16

17 Let α =(α 1,..., α n ), β =(β 1,...,β m ) and γ = γ ij,then the i=1,...,n, j=1,...,m analogue of eq. (10) is α = βγ (12) which is generally not satisfied unless we impose restriction on α, β and γ. 13 The following facts are easily verified : The equality (12) holds if and only if there is a probability distribution p on {(i A,j B ); i =1,..., n, j =1,..., m} whose marginal distribution of A and B are p A and p B respectively. In such a case p AjB is just the conditional probability p AjB (i A )=p (i A j B ) (13) and eq. (12) is then just the fundamental equation of conditional probability namely, for i =1,..., n p(i A )= X j p(j B )p(i A j B ) In the HSM framework the eq. (10) holds if and only if the observables A and B commute, we conclude that 1. Two commuting decision situations A and B canbemodelledby one standard probability space whose elementary events are the pairs (i A,j B ) of values of the two observables. 2. TwononcommutingDSsA and B cannot be modelled by a single probability space. In particular p AjB (i A ) is not a conditional probability in the formal sense since there is no underlying probability space in which the probabilities p (i A,j B ) is defined. Of course not all instances of non-commutativity in decision theory call for Hilbert space modelling. In particular, the standard modelling approach is satisfactory when the modification of the probability space following the first decision, can be expected. As an example of this kind, consider the 13 As an example take α = β = 1 2, 1 2 and γ = µ then γβ = 3 8, 5 8.

18 following two decision situations: A is a plan for the evening which involves a choice between going to a (long) opera play (1 A ) or staying at home (2 A ). The decision situation B is an afternoon party where the agent is invited to choose between drinking whisky (choice 1 B ) and drinking champagne (choice 2 B ). Assume that the probability for the direct choice are the following: p A (1 A )=0.3, p A (2 A )=0.7 while p B (1 B )=.5 =p B (2 B ). Let us further assume that we performed an experiment and observed frequencies of choice corresponding to the following probabilities: p A (1 A 1 B )= 0.1 and p A (1 A 2 B )=0.15. It is easy to verify that equation (10) is not satisfied. Indeed in this example there is no single probability model for which the probability p (1 A, 1 B ) can be defined.insuchacaseweclearlyseeagood reason why playing B before A modifies the probability distribution of the choices in A. Theexampleabove hardlyqualifies as a behavioral anomaly because we do expect preferences to be affected by the consumption of alcohol. Our point is that many instances of behavioral anomalies are very different from the example above. Those are instances in which we expect consistency with standard probabilistic model - because nothing justifies a modification of preferences - yet, actual behavior is inconsistent. TheHilbertspace model of preferences is useful when i) the conditional probability formula doesn t hold and ii) there is no good reason for why playing the intermediary DS would modify the probability distribution for the final DS. In such cases, a model in which the initial state (preference) is indeterminate may become more appealing. Situations characterized by i) and ii) are routinely encountered in experimental psychology or in preferences test. One example is provided by Erev, Bornstein and Wallsten (1993). They describe an experiment showing that revealed preferences are not consistent with the so called Information Reduction assumption (IR) which assumes that uncertainty can be reduced to a numerical probability. This assumption is implicitly made by most theorists including e.g. Savage (1954) but also by many decision theorists including Kahneman and Tversky (1979). One implication of IR is that subjects should make the same decisions regardless of whether they are asked to state their probabilities or not. In an experiment EBW (1993) compare the choices made by two groups one of which is invited to answer a questionnaire before playing a simple version of a n-person game called the intergroup public good The IPG game proceeds as follows. Each subject in the experiment is given a sum of 18

19 EBW found that asking subjects to fill in a questionnaire stating their subjective probabilities for others play did affect their subsequent choice. The revealed preferences in the two treatments were significantly different: In the group that played directly there was a positive correlation between the size of the expected price and the choice of investment (risky choice). While in the group that answered the questionnaire no such correlation existed. The authors propose that decision-makers asked to state probabilities or to give numerical assessments, pay less attention to the payoff dimension. We note that this interpretation is consistent with the assumption that the subjects are initially in a state of hesitation. The questionnaire operates on this state of mind, triggers a determination along a dimension correlated with the choices and therefore affects the distribution over final choices. This example may also be explained, using the HSM formalism, in terms of framing effect see section 3.3. We argue that if preferences change so easily (for instance under the sole influence of filling a questionnaire), a more attractive approach may be to view the agent as being in an indeterminate state. The Hilbert space formalism models this indeterminacy of preferences and provides a general framework to make predictions in this kind of situations. It allows for a unified treatment of situations consistent with stable innate preferences (when two observables can be modelled by a single probability space), and situations inconsistent with stable innate preferences. Most of the psychological explanations given for anomalies such as extremeness aversion, anchoring framing etc.. seem to be in line with our main idea that preferences are more appropriately modelled as being indeterminate and context dependent Experiment 1: superposition of states of mind Consider two identical populations of agents. Suppose that we can distinguish between two types of agents: altruist (a) or selfish (s). Assume that the type of the agent can be revealed by making him answer to a questionnaire. We propose an experiment where the two populations are invited to play a Prisoner Dilemma against a hidden player. Before playing the PD the agents from the first population are invited to answer the questionnaire. The agents money,5shekel,andchooseswhethertokeepitandcashinattheendofthesessionorto invest it. Members of the group with the higher number of investors receive a monetary reward of r Shekel each (In the experiment r was either 15 or 25 Shekels). In case of tie, members of both groups received r/2 units each. 19

20 from the second population play the PD game without any preparation. Assume that the experiment is made on the first population of agents and shows that the population consists of a proportion of α of altruist agents (and a proportion (1 α) of selfish agents). Denote by B the operator representing the questionnaire. Now, the agents are invited to play the Prisoner s Dilemma game (operator A) with the same payoff matrix as in the example above: 1 A (C) 2 A (D) [C] [D] Assume that we find that a proportion β of altruist agents choose to cooperate ((1 β) choose to defect) and a proportion γ of selfish agents choose to cooperate ((1 γ) to defect). The proportion of agents of the first population choosing C is thus p 1 (C) =αβ +(1 α)γ. The second part of the experiment is made on the other population of agents who play the PD game directly. From a classical point of view, agents are either altruist or selfish independently of the fact that they have answered the questionnaire (the types are fixed). Since the two populations are identical, we should be able to apply the conditional probability formula (although the magnitudes in the expression on the rhs of the equation below have not been measured for population 2) to obtain p 2 (C) =p(a)p(c a)+p(s)p(c s) =αβ +(1 α)γ. Assume that we find that the proportion of agents in population 2 who choose C is p 0 (C) which is significantly different from p 2 (C) above. The only way to explain such results within a classical framework would be to give up the conditional probability formula and hence to assume that the agents, though being all the time in a definite state (altruist or selfish) change the way they play the Prisoner s Dilemma game by the sole virtue of having answered the questionnaire. This assumption does not seem to be consistent with the idea of an agent having fixed determinate preferences. We propose that at the beginning, the agent is in a superposition of the agent is a superposition of two types: altruist and selfish. Both modes of 20

21 behavior are latent and have the potential to be observed. As the agent answers the questionaire, this hesitation is resolved: he has to choose beween the two behavioral inclinations (propensities to act). Before this operation, the two modes of behavior affect his choice (in the PD) but as soon as one mode has manifested itself, only this mode is active. If in fact the results with respect to the choice in the Prisoner s Dilemma turn out to be different in the two situations (having answered the questionnaire or not), the proposed experiment would provide support for the hypothesis that preferences are indeterminate i.e. that agents can be represented by a superposition of types Experiment 2: Order matters Consider three different DSs each with a choice between two alternatives: PD (Prisoner s Dilemma), DG (Dictator game) and UG where UG denotes the responder s decision situation in an ultimatum game where the first mover (a hidden player) proposes a given split of a known sum of money. The decision is whether to accept or to reject the split. In case of a rejection, the payoff is zero to both players, in particular to our decision maker - the responder. Suppose that we have empirically established that order matters for the following two pairs of DSs (PD, DG) and (DG,UG). According to our theory, these DSs must be represented by pairs of non-commuting operators. Then, unless the behavior types in the PD and UG are perfectly correlated i.e. that the proportion of players having chosen cooperate in the PD and choosing accept in the UG is either equal to 1 or to zero, our theory predicts that order matters in the play of the pair of DSs (PD,UG). 15 In order to test this prediction we propose the following experiment. We first establish whether or not order matters between the two first pairs of DSs (PD,DG) and (DG,UG) using the following procedure: We take two populations of identical individuals (assumed to be represented by the same state vector). The first population is invited to play the following sequence: first the PD (against a hidden player as in experiment 1) and thereafter 15 Technically speaking given three operators A, B, C (with two eigenvectors each), if both (A, B) and (B, C) are pairs of non-commuting operators, then the eigenvectors of A as well as those of C form a basis of the same two dimensional space. If in addition A and C commute then A and C must have the same eigenvectors. The two operators differ only by the names given to the eigenvalues. In other words the types in A are perfectly correlated with the types in C. 21

22 the DG (against a hidden third player). The individuals from the second population are invited to play the same DSs but in the opposite order: first the DG and thereafter the PD. If the distributions of choices turn out to be significantly different we have established that order matters when playing the PD and DG. We perform a similar test for (DG,UG) (with another set of participants). If we do observe that order matters even in this case, we move over to the next step. The next step of the experiment is to test whether or not the types in PD and UG are perfectly correlated and we do this by selecting a population of agents who have chosen cooperate in the PD. Assume that there is no evidence for perfect correlation we can then move to the last phase of the experiment namely testing the prediction that PD and UG are non-commuting decision situations. We do that by the same procedure we used to test the non commutatitvity of (PD,DG) and (DG,UG) (of course with another set of participants). If we do in fact observe that the distributions of choices are significantly different it will be consistent with our theory of preferences indeterminacy. Our objective is to take two examples of experiments where two different behavioral theories have been proposed e.g. the transfer effect for the WL and the PD games. Our contribution would be to link the two experimental phenomena with a cross prediction along the lines described above. If the prediction turns out to be correct but cannot be explained by either one of those theories, we would had demonstrated that our theory has the potential to both unify some of the existing body of behavioral economics and generate new results. 3.3 Framing Effects Providing a framework to think about framing effects is one of the interesting and promising application of the indeterminacy approach in decision theory. Kahneman and Tversky (2002 p. xiv) define the framing effectusingatwo steps(nonformal)modeloftheprocessofdecision-making. Thefirst step corresponds to the construction of a representation of the decision situation. The second to the evaluation of the choice alternatives. The crucial point is that the true objects of evaluation are neither objects in the real world or verbal description of those objects; they are mental representations. To capture this feature we propose that decision-making be modelled as a sequence of operators i.e. as a process whereby the agent is successively subjected 22

23 to two operators. Thus, we suggest to model frames as operators in a way similar to decision situations. A frame or FR is defined as a collection of alternative mental representations (of a decision situation). The corresponding operator is denoted A e where the tilde is used to distinguish operators associated with FRs from those associated with DSs. We model the process of constructing (selecting) a representation as the analogue of the measurement of the operator corresponding to the framing proposed to the agent i.e. the collapse onto one of the frame operators eigenvectors (onto one of the alternative mental representations). 16 In contrast with types associated with a DS, the types associated with a FR (hereafter we call them FR-type) are not directly observable but they could, in principle, be revealed by a questionnaire. The operators associated with FRs and DSs are defined within the same state space of representations/preferences. One interpretation is that representations and preferences have grown entangled through the activity of the mind i.e. decision processes. 17. We provide now an illustration of this approach. In Selten (1998) an experiment performed by Pruitt (1970) is discussed. Two groups of agents are invited to play a Prisoner Dilemma. The first group is presented the game in the usual matrix form but with the choices labelled 1 and 2 (instead of C and D presumably to avoid associations with the suggestive terms cooperate and defect ): 1 G (C) 2 G (D) [C] [D] The second group is presented the game with payoffs decomposed as 16 Interestingly, this approach is consistent with research in neurobiology that suggests that the perception of a situation involves operations in the brain very similar to those involved when chosing an action. To perceive is not only to combine, weight, it is to select. (p.10 Berthoz 2003) 17 In quantum physics the entanglement of the states of two spin 1/2 particles in the singlet spin state, means that the two particles form a single system that cannot be factored out into two subsystems even when separated by a great distance (cf. the famous Einstein Podolsky Rosen (EPR) non-locality paradox). 23

24 follows: For me For him 1 G G 1 0 The payoffs are the sums of what you take for yourself and what you get from the other player. Game theoretically it should make no difference whether the game is presented in the matrix form or in the decomposed form. However, an experiment made by Pruitt shows that one observes dramatically more cooperation in the decomposed form presentation. Selten proposes a bounded rationality explanation: the players would be making a superficial analysis and do not perceive the identity of the game presented in the two forms. We propose that the agents after having been confronted with the matrix presentation are not the same as before and not the same as the agents having been confronted with the decomposed form. The object of evaluation does not pre-exist the interaction between the agent and the framed decision situation. It is, as Kahneman and Tversky propose, constructed in the process of decision-making. Using the framework developed in this paper, we call A e the FR operator corresponding to the framing of the matrix presentation, B e the one corresponding to the framing of the decomposed presentation and G the DS operator associated with the game. G has two non-degenerated eigenvalues denoted 1 G and 2 G. The first step in the decision process corresponds to the selection of a mental representation of the decision situation. At that stage the agent is subjected to a frame A e or B. e Accordingly, before being presented the DS in some form, the agent is described by a superposition of possible mental representations. For the sake of illustration we assume that each frame only consists of two alternative E representations. E We can write ψi = α 1 ea 1 i + α 2 ea 2 i and ψi = β e 1 b1 + β e 2 b2. We propose the following description of the mental representations (this is only meant as a suggestive illustration) ea 1 i : decision situation perceived as an artificial small stake DS; ea 2 i : decision situation perceived as a real stake DS by analogy with real life PDE like situations (often occurring in a repeated setting). eb 1 : decision situation perceived as a test of generosity; E eb 2 : decision situation perceived as a test of smartness. 24